Probability Distributions and Variances of Quadratic Loss Functions

The use of quadratic loss functions has been advocated in quality engineering and experimental design for process optimization and robust design. We derive theoretical density functions and variances for nominal-the-best, smaller-the-better, and larger-the-better quadratic loss functions in general and when the response variable has a specified distribution. While considerable attention has been given to individual and mean loss, in some applications it is of interest also to know something about the loss distribution and variance. Results frequently exhibit high variance and skew and unique density functions in cases for which it is not advisable to base decisions on expected loss alone.